A speaks truth in 75% cases and B speaks truth in 80% cases. Find the probability that they contradict each other in stating the same fact?

Answer

To solve the problem of finding the probability that A and B contradict each other while stating the same fact, we can follow these steps:

### Step 1: Define the probabilities
Let:
- \( P(A) \) = Probability that A speaks the truth = 75% = \( \frac{75}{100} = \frac{3}{4} \)
- \( P(B) \) = Probability that B speaks the truth = 80% = \( \frac{80}{100} = \frac{4}{5} \)

### Step 2: Calculate the probabilities of lying
Now, we need to find the probabilities that A and B lie:
- Probability that A lies, \( P(A') = 1 - P(A) = 1 - \frac{3}{4} = \frac{1}{4} \)
- Probability that B lies, \( P(B') = 1 - P(B) = 1 - \frac{4}{5} = \frac{1}{5} \)

### Step 3: Determine the scenarios for contradiction
A and B will contradict each other in two scenarios:
1. A speaks the truth while B lies.
2. A lies while B speaks the truth.

### Step 4: Calculate the probabilities for each scenario
1. Probability that A speaks the truth and B lies:
\[ P(A \text{ speaks truth and } B \text{ lies}) = P(A) \times P(B') = \frac{3}{4} \times \frac{1}{5} = \frac{3}{20} \]

2. Probability that A lies and B speaks the truth:
\[ P(A' \text{ and } B \text{ speaks truth}) = P(A') \times P(B) = \frac{1}{4} \times \frac{4}{5} = \frac{1}{5} = \frac{4}{20} \]

### Step 5: Add the probabilities of the two scenarios
Now, we add the probabilities of the two scenarios to find the total probability that A and B contradict each other:
\[ P(\text{Contradiction}) = P(A \text{ speaks truth and } B \text{ lies}) + P(A' \text{ and } B \text{ speaks truth}) = \frac{3}{20} + \frac{4}{20} = \frac{7}{20} \]

### Final Answer
Thus, the probability that A and B contradict each other in stating the same fact is:
\[ \boxed{\frac{7}{20}} \]